Is there any way to measure angle between edges/faces, or change the angle so that the mesh changes according to the angle we input?
As well as the ruler and protractor option that was added in blender 2.67, you can also display edge length and angles while you are editing.

In 2.80+ you can find the length and angle options in the viewport overlay options menu.



For v2.7x, you can turn on these options under the Mesh Display panel of the properties region - N